mathematics of computation volume 63,number 208 october 1994, pages 717-725 PROOF OF A CONJECTURED ASYMPTOTIC EXPANSION FOR THE APPROXIMATION OF SURFACE INTEGRALS P. VERLINDEN AND R. COOLS Abstract. Georg introduced a new kind of trapezoidal rule and midpoint rule to approximate a surface integral over a curved triangular surface and conjec- tured the existence of an asymptotic expansion for this approximation as the subdivision of the surface gets finer. The purpose of this paper is to prove the conjecture. 1. Introduction In [1] Georg introduced a new kind of trapezoidal rule and midpoint rule to approximate a surface integral over a curved triangular surface and made a con- jecture about the asymptotic behavior of the approximation as the subdivision of the surface gets finer. Recently, Georg and Tausch [2] found a partial proof of this conjecture. Strong numerical evidence supports the validity of the con- jecture. The purpose of this paper is to prove the conjecture. Simultaneously and independently, Lyness [3] obtained similar results. Let a denote the triangle a :={{u,v)£ R2:0<u,0<v,u + v< 1} and V := {(0,0), (1,0), (0,1)} its vertices. Let af ¡ and af ¡ denote the affine maps n i \ (u+i V + j\ (I -U + i 1 -V + A alj{u,v):= (—,—iy alj{u,v):= (__,__¿J and tfn := {a?j : 0 <i + j< n - 1}U {äftj : 0 < /'+ j < n - 2}. Then e= \J aio) represents the regular subdivision of the triangle o into n2 subtriangles. Let (f> : a -> M3 : (w, v) i->(x, v, z) = (¡>{u, v) Received by the editor June 2, 1993 and, in revised form, December 7, 1993. 1991MathematicsSubjectClassification. Primary 65N38, 65D30. Key words and phrases. Numerical integration, surface integral, Euler-Maclaurin expansion, boundary element method. ©1994 American Mathematical Society 0025-5718/94 $1.00 + $.25 per page 717 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use